I will bet your problem is caused by the brakes. The first thing I would do is pull the hubs and look at the condition of the shoes, action of wheel cylinders, return springs, grease on drums,etc. If all that looks good the front brake shoes are probably out of adjustment. When you apply the brakes one set a brakes is grabing the hub at a slightly different time than the other. Try this method of adjustment on all 4 wheels, it has worked for me for years.
Jack the whole car up so you have all four wheels off the ground. Release the emergency brake. Wedge a 2 X 4 between the brake pedal and the seat so that you have a medium presure on the pedal. You may have to play with this until you get the feel of it. Now adjust the brakes on each wheel so that the drag on the wheel is the same on all 4 wheels. You will have to turn the wheel to determine the amount of drag you have, let off on the brake pedal, make your adjustment, then put your 2X4 back on the pedal and test your drag again. When you get all 4 wheel with the same drag, while the wheel turns freely without rubbing on your brake shoes, the brakes will be adjusted perfectly. I know that sounds a little undignified but this method has worked great for me for years.